Modeling the Transition between Localized and Extended Deposition in Flow Networks through Packings of Glass Beads Gess Kelly1 Navid Bizmark34 Bulbul Chakraborty1 Sujit S. Datta3 and Thomas G. Fai2

2025-05-06 0 0 7.37MB 10 页 10玖币

侵权投诉

Modeling the Transition between Localized and Extended Deposition in Flow

Networks through Packings of Glass Beads

Gess Kelly1, Navid Bizmark3,4, Bulbul Chakraborty1, Sujit S. Datta3, and Thomas G. Fai2∗

1Martin A. Fisher School of Physics, Brandeis University, Waltham, MA 02453

2Mathematics Department & Volen Center for Complex Systems, Brandeis University, Waltham, MA 02453

3Department of Chemical and Biological Engineering,

Princeton University, Princeton, NJ 08544, USA and

4Princeton Materials Institute, Princeton University, Princeton, NJ 08540, USA

(Dated: March 28, 2023)

We use a theoretical model to explore how ﬂuid dynamics, in particular, the pressure gradient and

wall shear stress in a channel, aﬀect the deposition of particles ﬂowing in a microﬂuidic network.

Experiments on transport of colloidal particles in pressure-driven systems of packed beads have

shown that at lower pressure drop, particles deposit locally at the inlet, while at higher pressure

drop, they deposit uniformly along the direction of ﬂow. We develop a mathematical model and use

agent-based simulations to capture these essential qualitative features observed in experiments. We

explore the deposition proﬁle over a two-dimensional phase diagram deﬁned in terms of the pressure

and shear stress threshold, and show that two distinct phases exist. We explain this apparent phase

transition by drawing an analogy to simple one-dimensional models of aggregation in which the

phase transition is calculated analytically.

I. INTRODUCTION

Deposition and aggregation of ﬁne particles in mi-

croﬂuidic networks and porous media play an impor-

tant role in various natural and industrial processes

such as water puriﬁcation, geotextile ﬁltration, appli-

cations in precision drug delivery and similar biomed-

ical tasks, transport of microplastics, environmental

cleanups, groundwater pollutant removal, oil recovery,

and transport of nanomaterials for groundwater aquifer

remediation [1–7] [8–11]. For example, in ﬁltration pro-

cesses, understanding of the deposition dynamics of col-

loidal particles plays a signiﬁcant role in improving ﬁlter

eﬃciency via reducing ﬁlter fouling [12–14]. Observa-

tions from [15] indicate that, regardless of the charge of

the colloidal particles ﬂowing in the bead network, apply-

ing lower pressures across the system leads to localized

deposition under various conditions. This may suggest

that irrespective of the exact local clogging mechanism

(e.g., bridging versus aggregation [16]), the interplay of

hydrodynamical variables in these systems controls the

resulting deposition proﬁle. We focus on the role of ap-

plied pressure diﬀerence ∆Pas one of the key variables

motivated by the experimental design in [15] and the wall

shear stress τw, which has been shown in past studies

to play an important role in erosion [17–19]. Here, the

shear stress at the wall τwrefers to the shear stress ex-

perienced by colloidal particles deposited on the walls of

the packing. We follow the approach of [19] to capture

the role of the shear stress threshold τ, a material pa-

rameter that describes the threshold shear stress at the

wall above which ﬂuid ﬂow erodes the deposited particles

from the walls. Table S1 in the Supplementary Material

∗tfai@brandeis.edu

contains representative parameter values. Throughout

the text, we use a hat notation, e.g., ∆ ˆ

P, to denote the

corresponding variables, e.g., ∆P, that are normalized

by a set value relevant to the experimental system. Ta-

ble S2 in the Supplementary Material contains additional

details.

Our speciﬁc system of interest is motivated by recent

experiments from [15], in which a constant pressure dif-

ference ∆Papplied to a packing of disordered glass beads

of length Ldrives a ﬂuid ﬂow containing a suspension of

colloidal particles. These experiments show that at larger

pressure diﬀerences, the proﬁle of particles deposited on

the solid matrix extends uniformly along the length of the

packing, while at lower pressures, the particles deposit lo-

cally at the inlet where they are injected into the system.

Here, we develop a mathematical model to explain how

the pressure diﬀerence inﬂuences the deposition proﬁle.

Past studies of simple mass-aggregation models [14, 21]

motivate us to explore the phase space of shear stress

threshold ˆτand pressure diﬀerence ∆ ˆ

P. In particular,

Majumdar et al. [21] consider minimal systems and lat-

tice models in which discrete masses diﬀuse at a constant

unit rate, which normalizes the overall timescale. Multi-

ple masses may aggregate at lattice sites after diﬀusion,

and units of masses erode (chip away) from blocks at

a constant chipping rate w. Physically, chipping corre-

sponds to single-particle dissociation in processes such as

polymerization and competes with coalescence. In this

simplest case, they work with two independent variables,

the chipping rate wand mass density ρ, that remain con-

stant and determine the behavior of the system at steady

state. They explore the phase space consisting of the

mass density ρand chipping rate wand show that these

ﬁnite systems exhibit two distinct phases at steady state,

only one of which involves an inﬁnite aggregate. One im-

portant diﬀerence between the simple mass-aggregation

model and our study is the ﬁxed density or constant to-

arXiv:2210.01780v2 [cond-mat.soft] 26 Mar 2023

FIG. 1. We use a network approach to model the bead packing

here shown in the absence of particles. (a) We skeletonize

the image of the packing, and then generate a network. The

edges of the network represent the channels through which

ﬂuid may ﬂow in the packing and the nodes represent the

junctions where these channels meet. (b) We obtain the ﬂow

rates in the channels by applying the Kirchhoﬀ’s laws [20].

color in (b) and (c) shows the magnitude of the channel ﬂow

rates in SI units (m3/s). (a) and (b) have the same scale bar.

The grey background shows the experimental micrograph of

the beads.

tal mass with periodic boundary conditions in contrast

to our model where there is a ﬂux of particles into and

out of the system.

We formulate the ﬂuid ﬂow through the packings by

applying the hydraulic analogy to the network of channels

extracted from the bead packing images. Using our net-

work model and deposition and erosion laws, we demon-

strate a similar transition in the normalized shear stress

threshold ˆτand pressure ∆ ˆ

Pphase space. Motivated by

these simple models of aggregation and fragmentation ex-

plored in previous studies [21, 22], we explore the model

phase space spanned by two dimensionless parameters,

and identify a transition between extended and localized

deposition regimes in terms of the key parameters of pres-

sure diﬀerence and shear stress threshold [23].

II. METHODS

We use a graph- or network-based approach [24, 25]

to model the porous network created by the beads as

shown in FIG. 1(a). The idea of modeling a porous sys-

tem as a network has been studied previously [26–28].

For instance, past studies have demonstrated the eﬀec-

tiveness of a network-based approach by highlighting the

role of disorder on the ﬂow distribution in porous media

[29]. We use images of two-dimensional (2D) slices of

the three-dimensional (3D) packing. We then generate

the model network based on these images. Because of

the expected diﬀerences between the ﬂow in 2D and 3D,

we do not expect to quantitatively recover all aspects of

the experiments. In such network models, each pore or

channel is typically represented by an edge in a network

representing the entire porous system (see FIG. 1.(a)).

Each edge may be weighted in terms of its conductance

and the nodes of the network represent junctions between

the edges. Assuming we have an incompressible ﬂuid, the

inﬂow and outﬂow of particles and ﬂuid must be equal

to respect mass conservation. In our system of interest,

boundary junctions at the inlet and outlet are subject to

two pressures held constant for the duration of the ex-

periment. To solve for the resulting channel ﬂow rates,

as shown in FIG. 1 (b) and (c), we apply Kirchhoﬀ’s

circuit laws. For each channel, we estimate the channel

length land diameter dfrom the image of the network to

calculate the channel conductance g, which is deﬁned as

the proportionality constant between the volumetric ﬂow

rate through a given channel and the pressure diﬀerence

across the channel given by the Hagen-Poiseuille law [20]:

g=πd4

128ηl ,(1)

where ηis the dynamic viscosity. The resolution of the

image in FIG. 1 tends to be lower along the boundaries

and our image processing does not accurately identify a

signiﬁcant portion of the channels. We use the largest

connected component of the model network, which is in

the interior of the packing. For this reason we neglect

the upper and lower boundaries. More details regard-

ing channel ﬂow rate calculations can be found in the

Supplementary Material. The total ﬂow rate is of order

10−10 m3/s once we account for the depth of the three-

dimensional system.

To capture the stochastic eﬀects, we use agent-based

modeling to model the particles as they deposit and erode

within the network. This distinguishes our study from a

closely related previous model of erosion in networks in

which diﬀerential equations are used to predict how ero-

sion changes the width of the channels in the network

[28]. Another diﬀerence is our assumption that the glass

beads that form the network remain ﬁxed over the course

of the simulation. Consequently, while the deposited par-

ticles may erode in our simulation, the channels them-

selves do not erode. Because initially the channel does

not contain any deposited particles, and since erosion

only occurs through removal of particles, the channel

width cannot grow beyond its initial value.

Particles enter the system from the inlet at constant

time intervals. This is a discrete approximation to the

experiments, in which the particles are injected continu-

ously at a constant volume fraction. This is also diﬀer-

ent from the conserved-mass aggregation models of [21]

where the density is constant. As particles deposit in the

network during the simulations, they cause a decrease in

the width of the channels, which may eventually lead to

topological changes when the number of deposited par-

ticles surpasses the channel capacity, i.e., clogging. We

assume that each time a particle is deposited (eroded),

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

ModelingtheTransitionbetweenLocalizedandExtendedDepositioninFlowNetworksthroughPackingsofGlassBeadsGessKelly1,NavidBizmark3;4,BulbulChakraborty1,SujitS.Datta3,andThomasG.Fai21MartinA.FisherSchoolofPhysics,BrandeisUniversity,Waltham,MA024532MathematicsDepartment&VolenCenterforComplexSystems,Brandeis...

展开>> 收起<<

Modeling the Transition between Localized and Extended Deposition in Flow Networks through Packings of Glass Beads Gess Kelly1 Navid Bizmark34 Bulbul Chakraborty1 Sujit S. Datta3 and Thomas G. Fai2.pdf

共10页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Modeling the Transition between Localized and Extended Deposition in Flow Networks through Packings of Glass Beads Gess Kelly1 Navid Bizmark34 Bulbul Chakraborty1 Sujit S. Datta3 and Thomas G. Fai2

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: